The Unique Factorisation Theorem
Every polynomial of degree n can be factorised into
linear factors, some of which may be repeated, so that if
is factorised as
then
The constant
and the roots
may be either real or complex. The factorisation is unique apart from the order of the factors. This means that the order
of each rootis well defined.
If we have two polynomials with the same factorisation apart from the order of the factors, then we can reorder the factors of one polynomial to match the order of the other. Expanding the brackets for both factorisations then returns identical polynomials, or cancelling brackets and powers, one at a time returns 1=1.
Every polynomialwhose coefficients
are all real can be expressed uniquely as a product of real linear and real quadratic factors. For any real quadratic factor
which cannot be written as a product of real linear factors, the discriminant
If
is a root of a polynomial with real coefficients then its complex conjugate 
is also a root and these roots exist for any polynomial whose factorisation into real factors contains the quadratic giving rise to the rootsand
Every polynomial of odd degree has at least one real zero, since we can factorise the polynomial either into three real linear factors, with each real linear factor returning one real root, or a real linear factor and a real quadratic, with the real linear factor giving one root and the quadratic returning complex conjugate roots.